Excel file: 7.PerfectSquares
Word document: 7.AlternativeVisionsOfPerfectSquares
This is a fun exercise that can be used as soon as students know about multiplication. This exercise shows a very slick way of counting sharpest isosceles triangles on odd regular polygons by showing the connection between the triangular image to dots organized into a perfect square array. The Excel file is a self-contained teaching tool which relies on a series of click-boxes that build out the model via a series of discussion points or questions on the Sharpest Triangles sheet. Once the image is built and triangles emanating from each apex are individually counted, we show how one can sum across apexes by considering a square array of dots on the Square sheet. This leads to a quick resolution of the problem. Two additional counting formulas are discussed in the Square sheet that will prove useful in analyzing more general images in Files 8, 9, and 10.
This is not the first piece I wrote with Don Chakerian but it is the one that can be explained all the way down to third grade. (We initially worked on counting sharpest triangles on even regular polygons because those images emerged from File 4. Only later did we discover that the odd cases had easier answers although they required a different way of creating the images. We will look at these images using Files 8 and 9.)